How to do summation or how to find another representation of the sum that runs over integer compositions. $$ \sum_{r_1+ \ldots + r_{L}= k} \left(-\frac{ a}{1+a} \right)^{k-L} n^{(3 r_1)} (n+3 r_{1}-1)^{(3 r_{2})}\cdot \ldots \cdot (n+\sum_{i=1}^{L-1} 3 r_i- L+1)^{(3 r_{L})} $$ where $\sum r_i = k$ runs over all integer compositions of $k$ and $L$ is a number of parts in the composition and $n^{(3 r_1)}$ is the rising factorial. $n$ is a positive integer or zero. I mean usual definition of compositions as in https://en.wikipedia.org/wiki/Composition_(combinatorics).

Realization in Mathematica may become somehow useful

```
Sum[(-(a/(1 + a)))^(p - r) Product[Pochhammer[n + Plus@@Table[3 k[[i]]-1, {i, 1, j - 1}],3 k[[j]]],{j, 1, r}],{r, 1, p},{k,Compositions[p-r,r]+1}]
```

`HeavisideTheta[j - 1/2]`

is always $1$ since your`j`

is always a natural number $\endgroup$11more comments